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Paradox of Material implication

Posted: Sun Sep 10, 2017 9:25 am
by Averroes
So we have the following deductive sequents.
$$1. \sim A \vdash A \supset B \hspace{5em} (PMI)$$
$$2.\ \ \ A \vdash B \supset A \hspace{5em} (PMI)$$

Intuitively, by reference to the truth table, you can understand that they are true. For the first one, when $\sim A$ is true, $A$ is false and $A \supset B$ is true. For the second, when $A$ is true, $B \supset A$ is true. But this is semantic entailment, i.e there is no interpretation where the premises are true and the conclusion false. In SD, we need to prove these using inference rules.

We can prove these as follows.

$1. \sim A \vdash A \supset B$



$1. \sim A \hspace{10em} Assumption$
$ \ \ \ 2. \ A \hspace{10em} Assumption$
$ \ \ \ \ \ \ 3. \sim B \hspace{8em} Assumption$
$ \ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$
$ \ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$
$ \ \ \ 6.\ B \hspace{10em} 3-5 \sim E$
$7. \ A \supset B \hspace{10em} 2,6 \supset I$

Compare with (the content is the same as above but different display):

Code: Select all

1. ~A                     Assunmption
    2.A                   Assumption
       3.~B              Assumption
       4.A                1 Reiteration
       5.~A                2 Reiteration
    6.B                3-5 ~E
7. A ⊃ B               2,6 ⊃I
______________

$2.\ A \vdash B \supset A$


$1.\ A \hspace{10em} Assumption$
$\ \ \ 2.\ B \hspace{9em} Assumption$
$\ \ \ 3.\ A \hspace{9em} 1,\ Reiteration$
$3.\ B \supset A \hspace{10em} 1,2 ⊃I$

Compare with (the content is the same thing as above):

Code: Select all

1. A                Assumption
    2. B             Assumption
    3. A              1 Reiteration
3.B ⊃ A            1,2 ⊃I
__________________

It is called LaTex notation. For example, the first deduction above looks like this in the post editor:

Code: Select all

$1. \sim A \hspace{10em} Assumption$
$ \ \ \ 2. \ A \hspace{10em} Assumption$
$ \ \ \ \ \ \ 3. \sim B \hspace{8em}  Assumption$
$ \ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$
$ \ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$
$ \ \ \ 6.\ B \hspace{10em} 3-5 \sim E$
$7. \ A \supset B \hspace{10em} 2,6 \supset I$
I have added functionality to the program of the forum such that when a poster type some specific codes within two dollar signs, an effect is produced when displayed. You can also display a lot of cool math stuffs, like matrices. Let us display a magic square.

$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$

Code: Select all

$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$
Later, you will be studying predicate logic. Such sentences as $\forall x \exists y (Lxy)$. It is so easy to write that now. It looks like this in the post editor:

Code: Select all

$\forall x \exists y (Lxy)$

The syntax of the language is shown here: https://www.abisource.com/~msevior/math-reference.html


One just has to put this syntax between a single pair of dollar signs or a double pair of dollar signs. Integrals, summations, greek alphabets, and all mathematical symbols can be displayed nicely with this language and functionality. What do you think, isn't this cool?

Re: Paradox of Material implication

Posted: Sun Sep 10, 2017 11:22 am
by Averroes
ProfAlexHartdegen wrote: Do you recommend any logic book for beginners that might include the paradox for material implication and the contraposition rule above.
I can do better than just recommend you! I can give you a link to a free digital copy! :)

It is on Scribd and you will have to sign up with Scribd to download it and read it. It is free anyway! However, it is not a true digital copy but rather a digitized photocopy of the paper book. It is here, Modern Logic by Graeme Forbes: https://www.scribd.com/doc/140431270/Gr ... dern-Logic

It is a very good introductory textbook for modern logic, and it mentions contraposition, converse (pg 21 in the book), and PMI (pg 123). You can read about expressive completeness that I mentioned on PhilosophyNow on pages 74-80. Also, the book teaches a proof format which the author refers to as the Lemmon format (proofs are introduced on page 86). This format is similar but different than the box format that you have so far studied. I recommend that you learn this format; it is easier to learn if you already know the box format. This book is a book that speaks to a wide range of audience, from philosophers through mathematicians to computer scientists.

Introductory texts, as in all fields, will teach you the alphabets, but it takes patience and practice to move from alphabets to writing meaningful essays and books. So take it easy and move at a pace such that you are benefiting from these resources. Much of the materials in this new book, you have already covered. Now, you take that which is new and revisit that which you already know. Just remember that you have already been processing these information, and now it will be a much lighter read. I am telling you this so that you do not feel discouraged when you see the thickness of the book ; 300 pages for classical logic (the one you are studying) and after that the author also considered some extensions to classical logic. For now focus on classical logic, once you get the mindset of a logician, the others will follow if you decide one day to go there.
Now I begin to see that although my book is a fairly good book for beginning students of Logic, it seems to omit some important logical rules and tools for solving proofs.
I tell you that even if you knew these rules, as a beginner it would have been difficult for you to find the answer for this one. In my humble opinion, it was a tough problem. I myself did not learn this proof before, I got an insight through thinking about the OP for sometime. Do not worry if you did not find the answer to this problem. The important thing for you now, is to get the mindset of a logician; and I think that you working hard on problems like these, by doing research and thinking by yourself, is already a significant part of this mindset! :)